Frozen Gaussian Approximation-Based Artificial Boundary Conditions for One-Dimensional Nonlinear Schrödinger Equation in the Semiclassical Regime

In this paper, we first analyze the stability of the unified approach, proposed in Zhang et al. (Phys Rev E 78:026709, 2008), for the nonlinear Schödinger equation in the semiclassical regime. The analysis shows that small semiclassical parameters deteriorate the accuracy of the unified approach, which will be also verified by numerical examples. Motivated by the time-splitting spectral method (Bao et al. in SIAM J Sci Comput 25:27–64, 2003), we generalize our previous work (Yang and Zhang in SIAM J Numer Anal 52:808–831, 2014), and propose frozen Gaussian approximation (FGA)-based artificial boundary conditions for solving one-dimensional nonlinear Schrödinger equation on unbounded domain. We split the linear part of the Schrödinger equation from the nonlinear part, and deal with the artificial boundary condition of the linear part by a simple strategy that all the Gaussian functions, whose dynamics are governed by the Hamiltonian flows, going out of the domain will be eliminated numerically. Since the nonlinear part is given by ordinary differential equations, it does not require artificial boundary conditions and can be solved directly. This strategy also works for the nonlinear Schrödinger equation with periodic lattice potential by using Bloch decomposition-based FGA. We present numerical examples to verify the performance of proposed numerical methods.